MathDB
Nonempty subsets and circle !

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May 16, 2015
combinatorics

Problem Statement

Let M={1,2,...,2013} M=\{1,2,...,2013\} and let Γ \Gamma be a circle. For every nonempty subset B B of the set M M , denote by S(B) S(B) sum of elements of the set B B , and define S(∅)=0 S(\varnothing)=0 ( ∅ \varnothing is the empty set ). Is it possible to join every subset B B of M M with some point A A on the circle Γ \Gamma so that following conditions are fulfilled:
1 1 . Different subsets are joined with different points;
2 2 . All joined points are vertices of a regular polygon;
3 3 . If A1,A2,...,Ak A_1,A_2,...,A_k are some of the joined points, k>2 k>2 , such that A1A2...Ak A_1A_2...A_k is a regular k−gon k-gon , then 2014 2014 divides S(B1)+S(B2)+...+S(Bk) S(B_1)+S(B_2)+...+S(B_k) ?
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